We construct a gauge-invariant approximation of the energy density of an interacting electron gas in the presence of a non-uniform magnetic field. The interaction is approximated by a slowly varying local potential. The magnetic field is a superposition of a constant fieldBand that due to a magnetic dipoleμ, thus making the energy density suitable for direct calculation of chemical-shift tensors of, for example, interacting closed-shell systems. Unlike a similar theory for the magnetic susceptibility, the field-dependent exchange energy does not diverge.